8.1: L’Hôpital’s Rule Actually, L’Hôpital’s Rule was developed by his teacher Johann Bernoulli. De l’Hôpital paid Bernoulli for private lessons, and then published the first Calculus book based on those lessons. Guillaume De l'Hôpital 1661 - 1704 Greg Kelly, Hanford High School, Richland, Washington
8.1: L’Hôpital’s Rule Johann Bernoulli 1667 - 1748
Consider: If we try to evaluate this by direct substitution, we get: Zero divided by zero can not be evaluated, and is an example of indeterminate form. In this case, we can evaluate this limit by factoring and canceling:
L’Hôpital’s Rule: * of the form 0/0 If is indeterminate*, then:
We can confirm L’Hôpital’s rule by working backwards, and using the definition of derivative:
Example: If it’s no longer indeterminate, then STOP! If we try to continue with L’Hôpital’s rule: which is wrong, wrong, wrong!
On the other hand, you can apply L’Hôpital’s rule as many times as necessary as long as the fraction is still indeterminate: (Rewritten in exponential form.) not
L’Hôpital’s rule can be used to evaluate other indeterminate forms besides . The following are also considered indeterminate: The first one, , can be evaluated just like .
1
Indeterminate Form (0): make it look like 0/0 or / This approaches This approaches
Some other examples:
Indeterminate Forms 1, 00, 0: use logarithms
Other examples: Ans: 1 Ans: 1
Indeterminate form : combine the terms This is indeterminate form Now it is in the form L’Hôpital’s rule applied once. Fractions cleared. Still
L’Hôpital again.
NOT Indeterminate Forms: